The repressive theocratic regime in Margaret Atwood's “The Handmaid's Tale” is Gilead, formally the Republic of Gilead.
Atwood's 1985 dystopian novel is set in a near-future New England after a radical movement overthrows the United States government. Gilead is a patriarchal, totalitarian theonomy that claims religious justification for its laws. It strips women of property, literacy, employment, and control over their own reproductive lives.
The narrator, Offred, is forced into the role of a Handmaid because she is fertile. Handmaids are assigned to elite Commanders and compelled to bear children under a state-sanctioned ritual called the Ceremony. Gilead's rigid social order also includes Wives, Marthas, Aunts, Econowives, and other controlled categories.
Mayday is the name of an underground resistance network, while the Sons of Jacob are the extremist group that helps establish Gilead. The Republic of Texas is not the regime in Atwood's novel, making “Gilead” the canonical answer.